where is the negative of the gravitational self-energy of the body (). This violation of the massive-body equivalence principle is known as the ``Nordtvedt effect''. The effect is absent in GR () but present in scalar-tensor theory (). The existence of the Nordtvedt effect does not violate the results of laboratory Eötvös experiments, since for laboratory-sized objects, , far below the sensitivity of current or future experiments. However, for astronomical bodies, may be significant ( for the Sun, for Jupiter, for the Earth, for the Moon). If the Nordtvedt effect is present () then the Earth should fall toward the Sun with a slightly different acceleration than the Moon. This perturbation in the Earth-Moon orbit leads to a polarization of the orbit that is directed toward the Sun as it moves around the Earth-Moon system, as seen from Earth. This polarization represents a perturbation in the Earth-Moon distance of the form
where and are the angular frequencies of the orbits of the Moon and Sun around the Earth (see TEGP 8.1  for detailed derivations and references; for improved calculations of the numerical coefficient, see [102, 53]).
Since August 1969, when the first successful acquisition was made of a laser signal reflected from the Apollo 11 retroreflector on the Moon, the lunar laser-ranging experiment (LURE) has made regular measurements of the round-trip travel times of laser pulses between a network of observatories and the lunar retroreflectors, with accuracies that are approaching 50 ps (1 cm). These measurements are fit using the method of least-squares to a theoretical model for the lunar motion that takes into account perturbations due to the Sun and the other planets, tidal interactions, and post-Newtonian gravitational effects. The predicted round-trip travel times between retroreflector and telescope also take into account the librations of the Moon, the orientation of the Earth, the location of the observatory, and atmospheric effects on the signal propagation. The ``Nordtvedt'' parameter along with several other important parameters of the model are then estimated in the least-squares method.
Several independent analyses of the data found no evidence, within experimental uncertainty, for the Nordtvedt effect (for recent results see [56, 154, 96]). Their results can be summarized by the bound . These results represent a limit on a possible violation of WEP for massive bodies of 5 parts in (compare Figure 1). For Brans-Dicke theory, these results force a lower limit on the coupling constant of 1000. Note that, at this level of precision, one cannot regard the results of lunar laser ranging as a ``clean'' test of SEP until one eliminates the possibility of a compensating violation of WEP for the two bodies, because the chemical compositions of the Earth and Moon differ in the relative fractions of iron and silicates. To this end, the Eöt-Wash group carried out an improved test of WEP for laboratory bodies whose chemical compositions mimic that of the Earth and Moon. The resulting bound of four parts in  reduces the ambiguity in the Lunar laser ranging bound, and establishes the firm limit on the universality of acceleration of gravitational binding energy at the level of .
In GR, the Nordtvedt effect vanishes; at the level of several centimeters and below, a number of non-null general relativistic effects should be present .
The most important such effects are variations and anisotropies in the locally-measured value of the gravitational constant, which lead to anomalous Earth tides and variations in the Earth's rotation rate; anomalous contributions to the orbital dynamics of planets and the Moon; self-accelerations of pulsars, and anomalous torques on the Sun that would cause its spin axis to be randomly oriented relative to the ecliptic (see TEGP 8.2, 8.3, 9.3 and 14.3 (c) ). An improved bound on of from the period derivatives of 20 millisecond pulsars was reported in ; improved bounds on were achieved using lunar laser ranging data , and using observations of the circular binary orbit of the pulsar J2317+1439 . Negative searches for these effects have produced strong constraints on the PPN parameters (Table 4).
Several observational constraints can be placed on using methods that include studies of the evolution of the Sun, observations of lunar occultations (including analyses of ancient eclipse data), lunar laser-ranging measurements, planetary radar-ranging measurements, and pulsar timing data. Laboratory experiments may one day lead to interesting limits (for review and references to past work see TEGP 8.4 and 14.3 (c) ). Recent results are shown in Table 5 .
Table 5: Constancy of the gravitational constant. For the pulsar data, the bounds are dependent upon the theory of gravity in the strong-field regime and on neutron star equation of state.
The best limits on still come from ranging measurements to the Viking landers and Lunar laser ranging measurements [56, 154, 96]. It has been suggested that radar observations of a Mercury orbiter over a two-year mission (30 cm accuracy in range) could yield .
Although bounds on from solar-system measurements can be correctly obtained in a phenomenological manner through the simple expedient of replacing G by in Newton's equations of motion, the same does not hold true for pulsar and binary pulsar timing measurements. The reason is that, in theories of gravity that violate SEP, such as scalar-tensor theories, the ``mass'' and moment of inertia of a gravitationally bound body may vary with variation in G . Because neutron stars are highly relativistic, the fractional variation in these quantities can be comparable to , the precise variation depending both on the equation of state of neutron star matter and on the theory of gravity in the strong-field regime. The variation in the moment of inertia affects the spin rate of the pulsar, while the variation in the mass can affect the orbital period in a manner that can subtract from the direct effect of a variation in G, given by . Thus, the bounds quoted in Table 5 for the binary pulsar PSR 1913+16  and the pulsar PSR 0655+64  are theory-dependent and must be treated as merely suggestive.
|The Confrontation between General Relativity and
Clifford M. Will
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